Hands-on_Ex4: Geographically Weighted Regression

Published

December 8, 2022

Overview

Geographically weighted regression is a spatial statistical technique that takes non-stationary variables into consideration and models the local relationships between these independent variables and an outcome of interest. In this hands-on exercise, I am going to build hedonic pricing models by using GWR methods. The dependent variable is the resale prices of condominium in 2015. The independent variables are divided into structural and locational.

The data

Two datasets will be used in this model.

  • Geospatial data: URA Master Plan subzone boundary in shapefile format

  • Aspatial data: condo_resale_2015 in csv format

Getting Started

The code chunk below is used to load related R packages into the environment.

pacman::p_load(olsrr, corrplot, ggpubr, sf, spdep, GWmodel, tmap, tidyverse, gtsummary)

Geospatial Data Wrangling

Importing geospatial data

mpsz <- st_read(dsn = "data/geospatial", layer = "MP14_SUBZONE_WEB_PL")
Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `H:\jocelyn276\ISSS624\Hands-on_Ex4\data\geospatial' using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

Updating CRS information

The code chunk below is used to update the EPSF code 3414.

mpsz_svy21 <- st_transform(mpsz, 3414)

The code chunk below is used to check the transform correct or not.

st_crs(mpsz_svy21)
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]

Aspatial data Wrangling

Importing aspatial data

condo_resale <- read_csv("data/aspatial/Condo_resale_2015.csv")
Rows: 1436 Columns: 23
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl (23): LATITUDE, LONGITUDE, POSTCODE, SELLING_PRICE, AREA_SQM, AGE, PROX_...

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

The code chunk below is used to display the data structure of the newly-imported file.

glimpse(condo_resale)
Rows: 1,436
Columns: 23
$ LATITUDE             <dbl> 1.287145, 1.328698, 1.313727, 1.308563, 1.321437,…
$ LONGITUDE            <dbl> 103.7802, 103.8123, 103.7971, 103.8247, 103.9505,…
$ POSTCODE             <dbl> 118635, 288420, 267833, 258380, 467169, 466472, 3…
$ SELLING_PRICE        <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1320…
$ AREA_SQM             <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 168,…
$ AGE                  <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22, 6,…
$ PROX_CBD             <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783402…
$ PROX_CHILDCARE       <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543, 0…
$ PROX_ELDERLYCARE     <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.121…
$ PROX_URA_GROWTH_AREA <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.410632,…
$ PROX_HAWKER_MARKET   <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969, 0…
$ PROX_KINDERGARTEN    <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076, 0…
$ PROX_MRT             <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.528…
$ PROX_PARK            <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.116…
$ PROX_PRIMARY_SCH     <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.709…
$ PROX_TOP_PRIMARY_SCH <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.709…
$ PROX_SHOPPING_MALL   <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.307…
$ PROX_SUPERMARKET     <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.581…
$ PROX_BUS_STOP        <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340, 0…
$ NO_Of_UNITS          <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34, 3…
$ FAMILY_FRIENDLY      <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0…
$ FREEHOLD             <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR       <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…

Converting aspatial data frame into a sf object

st_as_sf of sf package can be used to convert the tibble data frame to a sf object.

condo_resale.sf <- st_as_sf(condo_resale,
                            coords = c("LONGITUDE", "LATITUDE"),
                            crs=4326) %>%
  st_transform(crs=3414)

We can take a look at the sf object.

head(condo_resale.sf)
Simple feature collection with 6 features and 21 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 22085.12 ymin: 29951.54 xmax: 41042.56 ymax: 34546.2
Projected CRS: SVY21 / Singapore TM
# A tibble: 6 × 22
  POSTCODE SELLI…¹ AREA_…²   AGE PROX_…³ PROX_…⁴ PROX_…⁵ PROX_…⁶ PROX_…⁷ PROX_…⁸
     <dbl>   <dbl>   <dbl> <dbl>   <dbl>   <dbl>   <dbl>   <dbl>   <dbl>   <dbl>
1   118635 3000000     309    30    7.94   0.166   2.52     6.62   1.77   0.0584
2   288420 3880000     290    32    6.61   0.280   1.93     7.51   0.545  0.616 
3   267833 3325000     248    33    6.90   0.429   0.502    6.46   0.378  0.141 
4   258380 4250000     127     7    4.04   0.395   1.99     4.91   1.68   0.382 
5   467169 1400000     145    28   11.8    0.119   1.12     6.41   0.565  0.461 
6   466472 1320000     139    22   10.3    0.125   0.789    5.09   0.781  0.0994
# … with 12 more variables: PROX_MRT <dbl>, PROX_PARK <dbl>,
#   PROX_PRIMARY_SCH <dbl>, PROX_TOP_PRIMARY_SCH <dbl>,
#   PROX_SHOPPING_MALL <dbl>, PROX_SUPERMARKET <dbl>, PROX_BUS_STOP <dbl>,
#   NO_Of_UNITS <dbl>, FAMILY_FRIENDLY <dbl>, FREEHOLD <dbl>,
#   LEASEHOLD_99YR <dbl>, geometry <POINT [m]>, and abbreviated variable names
#   ¹​SELLING_PRICE, ²​AREA_SQM, ³​PROX_CBD, ⁴​PROX_CHILDCARE, ⁵​PROX_ELDERLYCARE,
#   ⁶​PROX_URA_GROWTH_AREA, ⁷​PROX_HAWKER_MARKET, ⁸​PROX_KINDERGARTEN

Exploratory Data Analysis

EDA using statistical graphics

The code chunk below is used to plot the distribution of SELLING_PRICE.

ggplot(data=condo_resale.sf, aes(x=`SELLING_PRICE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

We can find that the distribution of SELLING_PRICE is right-skewed, which means that more condominium units were transacted at relatively lower price.

We can use log transformation to normalise the skewed distribution. The code chunk below is used to derive a new variable called LOG_SELLING_PRICE.

condo_resale.sf <- condo_resale.sf %>%
  mutate(`LOG_SELLING_PRICE`=log(SELLING_PRICE))
ggplot(data=condo_resale.sf, aes(x=`LOG_SELLING_PRICE`)) +
  geom_histogram(bins=20, color="black", fill="light blue")

Multiple Histogram Plots distribution of variables

In this section, I am going to draw trellis plot by using ggarrange of ggpubr package.

The code chunk below is going to create 12 histograms.

AREA_SQM <- ggplot(data=condo_resale.sf, aes(x= `AREA_SQM`)) + 
  geom_histogram(bins=20, color="black", fill="light blue") +
  theme_bw(base_size=6)

AGE <- ggplot(data=condo_resale.sf, aes(x= `AGE`)) +
  geom_histogram(bins=20, color="black", fill="light blue") +
  theme_bw(base_size=6)

PROX_CBD <- ggplot(data=condo_resale.sf, aes(x= `PROX_CBD`)) +
  geom_histogram(bins=20, color="black", fill="light blue") +
  theme_bw(base_size=6)

PROX_CHILDCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_CHILDCARE`)) + 
  geom_histogram(bins=20, color="black", fill="light blue") +
  theme_bw(base_size=6)

PROX_ELDERLYCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_ELDERLYCARE`)) +
  geom_histogram(bins=20, color="black", fill="light blue") +
  theme_bw(base_size=6)

PROX_URA_GROWTH_AREA <- ggplot(data=condo_resale.sf, 
                               aes(x= `PROX_URA_GROWTH_AREA`)) +
  geom_histogram(bins=20, color="black", fill="light blue") +
  theme_bw(base_size=6)

PROX_HAWKER_MARKET <- ggplot(data=condo_resale.sf, aes(x= `PROX_HAWKER_MARKET`)) +
  geom_histogram(bins=20, color="black", fill="light blue") +
  theme_bw(base_size=6)

PROX_KINDERGARTEN <- ggplot(data=condo_resale.sf, aes(x= `PROX_KINDERGARTEN`)) +
  geom_histogram(bins=20, color="black", fill="light blue") +
  theme_bw(base_size=6)

PROX_MRT <- ggplot(data=condo_resale.sf, aes(x= `PROX_MRT`)) +
  geom_histogram(bins=20, color="black", fill="light blue") +
  theme_bw(base_size=6)

PROX_PARK <- ggplot(data=condo_resale.sf, aes(x= `PROX_PARK`)) +
  geom_histogram(bins=20, color="black", fill="light blue") +
  theme_bw(base_size=6)

PROX_PRIMARY_SCH <- ggplot(data=condo_resale.sf, aes(x= `PROX_PRIMARY_SCH`)) +
  geom_histogram(bins=20, color="black", fill="light blue") +
  theme_bw(base_size=6)

PROX_TOP_PRIMARY_SCH <- ggplot(data=condo_resale.sf, 
                               aes(x= `PROX_TOP_PRIMARY_SCH`)) +
  geom_histogram(bins=20, color="black", fill="light blue") +
  theme_bw(base_size=6)

ggarrange(AREA_SQM, AGE, PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, 
          PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN, PROX_MRT,
          PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH,  
          ncol = 3, nrow = 4)

Drawing Statistical Point Map

In this section, we are going to reveal the geospatial distribution condonimiumr resale prices in Singapore.

Firstly, we are going to turn on the interactive mode of tmap.

tmap_mode("view")
tmap mode set to interactive viewing

Next, the code chunk below is used to create an interactive point symbol map.

tm_shape(mpsz_svy21)+
  tmap_options(check.and.fix = TRUE) +
  tm_polygons()+
tm_shape(condo_resale.sf) +  
  tm_dots(col = "SELLING_PRICE",
          alpha = 0.6,
          style="quantile")
Warning: The shape mpsz_svy21 is invalid (after reprojection). See
sf::st_is_valid

The code chunk below is used to turn R display into plot mode.

tmap_mode("plot")
tmap mode set to plotting

Hedonic Pricing Modelling in R

Simple Linear Regression Method

First, we are going to build a simple linear regression model by using SELLING_PRICE as the dependent variable and AREA_SQM as the independent variable.

condo.slr <-  lm(formula=SELLING_PRICE~AREA_SQM, data = condo_resale.sf)
summary(condo.slr)

Call:
lm(formula = SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)

Residuals:
     Min       1Q   Median       3Q      Max 
-3695815  -391764   -87517   258900 13503875 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -258121.1    63517.2  -4.064 5.09e-05 ***
AREA_SQM      14719.0      428.1  34.381  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 942700 on 1434 degrees of freedom
Multiple R-squared:  0.4518,    Adjusted R-squared:  0.4515 
F-statistic:  1182 on 1 and 1434 DF,  p-value: < 2.2e-16

The summary result shows that the SELLING_PRICE can be explained by the formula:

y= 14719*x - 258121.1

The R-Squared is 0.4518, which shows that 45% of selling price can be explained by the regression model.

Since the p-value is much smaller than 0.0001, we can reject the null hypothesis that mean is a good estimator of SELLING_PRICE.

To visualise the best fit curve on a scatterplot, we can incorporate lm() as a method function in ggplot’s geometry as shown in the code chunk.

ggplot(data=condo_resale.sf,  
       aes(x=`AREA_SQM`, y=`SELLING_PRICE`)) +
  geom_point() +
  geom_smooth(method = lm)
`geom_smooth()` using formula = 'y ~ x'

Multiple Linear Regression Method

Visualising the relationships of the independent variables

Before building a multiple regression model, it is important to ensure that the independent variables used are not highly correlated to each other, to avoid multicollinearity.

Correlation matrix is commonly used to visualise the relationships between the independent variables. The code chunk below is used to plot the scatterplot to display the relatinship between the independent variables in condo_resale dataframe.

corrplot(cor(condo_resale[, 5:23]), diag = FALSE, order = "AOE",
         tl.pos = "td", tl.cex = 0.5, method = "number", type = "upper",
         number.cex = 0.3)

According to the scatterplot, it is clear that FREEHOLD is highly correlated to LEASE_99YEAR. It is wise to excluse LEASE_99YEAR.

Building a hedonic pricing model using multiple linear regression method

The code chunk below is used to calibrate the multiple linear regression model.

condo.mlr <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE    + 
                  PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
                  PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + PROX_KINDERGARTEN + 
                  PROX_MRT  + PROX_PARK + PROX_PRIMARY_SCH + 
                  PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + 
                  PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                data=condo_resale.sf)
summary(condo.mlr)

Call:
lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + 
    PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + 
    PROX_KINDERGARTEN + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + 
    PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sf)

Residuals:
     Min       1Q   Median       3Q      Max 
-3475964  -293923   -23069   241043 12260381 

Coefficients:
                       Estimate Std. Error t value Pr(>|t|)    
(Intercept)           481728.40  121441.01   3.967 7.65e-05 ***
AREA_SQM               12708.32     369.59  34.385  < 2e-16 ***
AGE                   -24440.82    2763.16  -8.845  < 2e-16 ***
PROX_CBD              -78669.78    6768.97 -11.622  < 2e-16 ***
PROX_CHILDCARE       -351617.91  109467.25  -3.212  0.00135 ** 
PROX_ELDERLYCARE      171029.42   42110.51   4.061 5.14e-05 ***
PROX_URA_GROWTH_AREA   38474.53   12523.57   3.072  0.00217 ** 
PROX_HAWKER_MARKET     23746.10   29299.76   0.810  0.41782    
PROX_KINDERGARTEN     147468.99   82668.87   1.784  0.07466 .  
PROX_MRT             -314599.68   57947.44  -5.429 6.66e-08 ***
PROX_PARK             563280.50   66551.68   8.464  < 2e-16 ***
PROX_PRIMARY_SCH      180186.08   65237.95   2.762  0.00582 ** 
PROX_TOP_PRIMARY_SCH    2280.04   20410.43   0.112  0.91107    
PROX_SHOPPING_MALL   -206604.06   42840.60  -4.823 1.57e-06 ***
PROX_SUPERMARKET      -44991.80   77082.64  -0.584  0.55953    
PROX_BUS_STOP         683121.35  138353.28   4.938 8.85e-07 ***
NO_Of_UNITS             -231.18      89.03  -2.597  0.00951 ** 
FAMILY_FRIENDLY       140340.77   47020.55   2.985  0.00289 ** 
FREEHOLD              359913.01   49220.22   7.312 4.38e-13 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 755800 on 1417 degrees of freedom
Multiple R-squared:  0.6518,    Adjusted R-squared:  0.6474 
F-statistic: 147.4 on 18 and 1417 DF,  p-value: < 2.2e-16

Preparing Publication Quality Table: ols method

According to the report above, it is not all the independent variables are statistically significant. We need to revise the model by removing the insignificant variables.

condo.mlr1 <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                   PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
                   PROX_URA_GROWTH_AREA + PROX_MRT  + PROX_PARK + 
                   PROX_PRIMARY_SCH + PROX_SHOPPING_MALL    + PROX_BUS_STOP + 
                   NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
                 data=condo_resale.sf)
ols_regress(condo.mlr1)
                             Model Summary                               
------------------------------------------------------------------------
R                       0.807       RMSE                     755957.289 
R-Squared               0.651       Coef. Var                    43.168 
Adj. R-Squared          0.647       MSE                571471422208.591 
Pred R-Squared          0.638       MAE                      414819.628 
------------------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 

                                     ANOVA                                       
--------------------------------------------------------------------------------
                    Sum of                                                      
                   Squares          DF         Mean Square       F         Sig. 
--------------------------------------------------------------------------------
Regression    1.512586e+15          14        1.080418e+14    189.059    0.0000 
Residual      8.120609e+14        1421    571471422208.591                      
Total         2.324647e+15        1435                                          
--------------------------------------------------------------------------------

                                               Parameter Estimates                                                
-----------------------------------------------------------------------------------------------------------------
               model           Beta    Std. Error    Std. Beta       t        Sig           lower          upper 
-----------------------------------------------------------------------------------------------------------------
         (Intercept)     527633.222    108183.223                   4.877    0.000     315417.244     739849.200 
            AREA_SQM      12777.523       367.479        0.584     34.771    0.000      12056.663      13498.382 
                 AGE     -24687.739      2754.845       -0.167     -8.962    0.000     -30091.739     -19283.740 
            PROX_CBD     -77131.323      5763.125       -0.263    -13.384    0.000     -88436.469     -65826.176 
      PROX_CHILDCARE    -318472.751    107959.512       -0.084     -2.950    0.003    -530249.889    -106695.613 
    PROX_ELDERLYCARE     185575.623     39901.864        0.090      4.651    0.000     107302.737     263848.510 
PROX_URA_GROWTH_AREA      39163.254     11754.829        0.060      3.332    0.001      16104.571      62221.936 
            PROX_MRT    -294745.107     56916.367       -0.112     -5.179    0.000    -406394.234    -183095.980 
           PROX_PARK     570504.807     65507.029        0.150      8.709    0.000     442003.938     699005.677 
    PROX_PRIMARY_SCH     159856.136     60234.599        0.062      2.654    0.008      41697.849     278014.424 
  PROX_SHOPPING_MALL    -220947.251     36561.832       -0.115     -6.043    0.000    -292668.213    -149226.288 
       PROX_BUS_STOP     682482.221    134513.243        0.134      5.074    0.000     418616.359     946348.082 
         NO_Of_UNITS       -245.480        87.947       -0.053     -2.791    0.005       -418.000        -72.961 
     FAMILY_FRIENDLY     146307.576     46893.021        0.057      3.120    0.002      54320.593     238294.560 
            FREEHOLD     350599.812     48506.485        0.136      7.228    0.000     255447.802     445751.821 
-----------------------------------------------------------------------------------------------------------------

Preparing Publication Quality Table: gtsummary method

The gtsummary package provides an elegant and flexible way to create publication-ready summary tables in R.

tbl_regression(condo.mlr1, intercept = TRUE)
Characteristic Beta 95% CI1 p-value
(Intercept) 527,633 315,417, 739,849 <0.001
AREA_SQM 12,778 12,057, 13,498 <0.001
AGE -24,688 -30,092, -19,284 <0.001
PROX_CBD -77,131 -88,436, -65,826 <0.001
PROX_CHILDCARE -318,473 -530,250, -106,696 0.003
PROX_ELDERLYCARE 185,576 107,303, 263,849 <0.001
PROX_URA_GROWTH_AREA 39,163 16,105, 62,222 <0.001
PROX_MRT -294,745 -406,394, -183,096 <0.001
PROX_PARK 570,505 442,004, 699,006 <0.001
PROX_PRIMARY_SCH 159,856 41,698, 278,014 0.008
PROX_SHOPPING_MALL -220,947 -292,668, -149,226 <0.001
PROX_BUS_STOP 682,482 418,616, 946,348 <0.001
NO_Of_UNITS -245 -418, -73 0.005
FAMILY_FRIENDLY 146,308 54,321, 238,295 0.002
FREEHOLD 350,600 255,448, 445,752 <0.001
1 CI = Confidence Interval
tbl_regression(condo.mlr1, 
               intercept = TRUE) %>% 
  add_glance_source_note(
    label = list(sigma ~ "\U03C3"),
    include = c(r.squared, adj.r.squared, 
                AIC, statistic,
                p.value, sigma))
Characteristic Beta 95% CI1 p-value
(Intercept) 527,633 315,417, 739,849 <0.001
AREA_SQM 12,778 12,057, 13,498 <0.001
AGE -24,688 -30,092, -19,284 <0.001
PROX_CBD -77,131 -88,436, -65,826 <0.001
PROX_CHILDCARE -318,473 -530,250, -106,696 0.003
PROX_ELDERLYCARE 185,576 107,303, 263,849 <0.001
PROX_URA_GROWTH_AREA 39,163 16,105, 62,222 <0.001
PROX_MRT -294,745 -406,394, -183,096 <0.001
PROX_PARK 570,505 442,004, 699,006 <0.001
PROX_PRIMARY_SCH 159,856 41,698, 278,014 0.008
PROX_SHOPPING_MALL -220,947 -292,668, -149,226 <0.001
PROX_BUS_STOP 682,482 418,616, 946,348 <0.001
NO_Of_UNITS -245 -418, -73 0.005
FAMILY_FRIENDLY 146,308 54,321, 238,295 0.002
FREEHOLD 350,600 255,448, 445,752 <0.001
R² = 0.651; Adjusted R² = 0.647; AIC = 42,967; Statistic = 189; p-value = <0.001; σ = 755,957
1 CI = Confidence Interval

Checking for multicolinearity

In the code chunk below, the ols_vif_tol() of olsrr package is used to test if there are sign of multicoliearity.

ols_vif_tol(condo.mlr1)
              Variables Tolerance      VIF
1              AREA_SQM 0.8728554 1.145665
2                   AGE 0.7071275 1.414172
3              PROX_CBD 0.6356147 1.573280
4        PROX_CHILDCARE 0.3066019 3.261559
5      PROX_ELDERLYCARE 0.6598479 1.515501
6  PROX_URA_GROWTH_AREA 0.7510311 1.331503
7              PROX_MRT 0.5236090 1.909822
8             PROX_PARK 0.8279261 1.207837
9      PROX_PRIMARY_SCH 0.4524628 2.210126
10   PROX_SHOPPING_MALL 0.6738795 1.483945
11        PROX_BUS_STOP 0.3514118 2.845664
12          NO_Of_UNITS 0.6901036 1.449058
13      FAMILY_FRIENDLY 0.7244157 1.380423
14             FREEHOLD 0.6931163 1.442759

Testing for Non-Linearity

In the code trunk below, the ols_plot_resid_fit() is used to perform linearity assumption test.

ols_plot_resid_fit(condo.mlr1)

Testing for Normality Assumption

ols_plot_resid_hist(condo.mlr1)

The code chunk is used to perform formal statistical test methods.

ols_test_normality(condo.mlr1)
Warning in ks.test.default(y, "pnorm", mean(y), sd(y)): ties should not be
present for the Kolmogorov-Smirnov test
-----------------------------------------------
       Test             Statistic       pvalue  
-----------------------------------------------
Shapiro-Wilk              0.6856         0.0000 
Kolmogorov-Smirnov        0.1366         0.0000 
Cramer-von Mises         121.0768        0.0000 
Anderson-Darling         67.9551         0.0000 
-----------------------------------------------

The summary table above reveals that the p-value of the four test are way smaller than the alpha value of 0.05. Hence we have the statistical evidence to reject the null hypothesis that the residual is NOT resemble normal distribution.

Testing for Spatial Autocorrelation

In order to perform spatial autocorrelation test, we need to convert condo_resale.sf from sf data frame into a SpatialPointDataFrame.

Firstly, we are going to export the residual of the hedonic pricing model and save it as data frame.

mlr.output <- as.data.frame(condo.mlr1$residuals)

Next, we are going to join the newly created data frame with condo_resale.sf object.

condo_resale.res.sf <- cbind(condo_resale.sf, 
                        condo.mlr1$residuals) %>%
rename(`MLR_RES` = `condo.mlr1.residuals`)

Then, we will convert condo_resale.res.sf from simple feature object into a SpatialPointsDataFrame because spdep package can only process sp conformed spatial data objects.

The code chunk below is used to perform the data conversion process.

condo_resale.sp <- as_Spatial(condo_resale.res.sf)
condo_resale.sp
class       : SpatialPointsDataFrame 
features    : 1436 
extent      : 14940.85, 43352.45, 24765.67, 48382.81  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 23
names       : POSTCODE, SELLING_PRICE, AREA_SQM, AGE,    PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN,    PROX_MRT,   PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH, PROX_SHOPPING_MALL, ... 
min values  :    18965,        540000,       34,   0, 0.386916393,    0.004927023,      0.054508623,          0.214539508,        0.051817113,       0.004927023, 0.052779424, 0.029064164,      0.077106132,          0.077106132,                  0, ... 
max values  :   828833,       1.8e+07,      619,  37, 19.18042832,     3.46572633,      3.949157205,           9.15540001,        5.374348075,       2.229045366,  3.48037319,  2.16104919,      3.928989144,          6.748192062,        3.477433767, ... 
tmap_mode("view")
tmap mode set to interactive viewing
tm_shape(mpsz_svy21)+
  tmap_options(check.and.fix = TRUE) +
  tm_polygons(alpha = 0.4) +
tm_shape(condo_resale.res.sf) +  
  tm_dots(col = "MLR_RES",
          alpha = 0.6,
          style="quantile") +
  tm_view(set.zoom.limits = c(11,14))
Warning: The shape mpsz_svy21 is invalid (after reprojection). See
sf::st_is_valid
Variable(s) "MLR_RES" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.

Switch back to “plot” mode.

tmap_mode("plot")
tmap mode set to plotting